5 
063 


-CONSTRUCTIVE   GEOMETRY 


E   R.  HEDRICK 


NEW  YORK 

THE  MACMILLAN  COMPANY 

1916 


CONSTRUCTIVE   GEOMETRY 


A   SERIES    OF   MATHEMATICAL   TEXTS 

EDITED    BY 

EARLE  RAYMOND  HEDRICK 


THE  CALCULUS 

By    ELLERY   WILLIAMS    DAVIS    and   WILLIAM    CHARLES 
BRENKE. 

PLANE   AND    SOLID    ANALYTIC    GEOMETRY 
By  ALEXANDER  ZIWET  and  Louis  ALLEN  HOPKINS. 

PLANE     AND     SPHERICAL     TRIGONOMETRY    WITH 

COMPLETE   TABLES 
By  ARTHUR  MONROE  KENYON  and  Louis  INGOLD. 

PLANE     AND     SPHERICAL    TRIGONOMETRY     WITH 

BRIEF   TABLES 
By  ARTHUR  MONROE  KENYON  and  Louis  INGOLD. 

THE   MACMILLAN   TABLES 

Prepared  under  the  direction  of  EARLE  RAYMOND  HEDRICK. 

PLANE   GEOMETRY 

By  WALTER  BURTON  FORD  and  CHARLES  AMMERMAN. 

PLANE   AND    SOLID   GEOMETRY 

By  WALTER  BURTON  FORD  and  CHARLES  AMMERMAN. 

SOLID   GEOMETRY 

By  WALTER  BURTON  FORD  and  CHARLES  AMMERMAN. 


CONSTRUCTIVE  GEOMETRY 


EXERCISES    IN    ELEMENTARY  GEOMETRIC 

DRAWING 


PREPARED   UNDER   THE   DIRECTION 

OF 

EARLE    RAYMOND    HEDRICK 


fforfe 
THE    MACMILLAN    COMPANY 

1916 

All  rights  reserved 


COPYRIGHT,  1916, 
BY  THE   MACMILLAN   COMPANY. 


Set  up  and  electrotyped.     Published  February,  1916. 


Nortooofi 

J.  S.  Gushing  Co. —  Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


PREFACE 

THE  saying  is  trite  that  students  who  enter  formal  courses  in  Euclidean  Geometry 
have  to  learn  both  the  strange  methods  of  formal  logic  and  the  equally  strange  geo- 
metric forms. 

A  course  to  acquaint  students  with  the  elementary  forms  and  constructions  is 
valuable  particularly  to  those  who  never  go  on  to  a  more  formal  course,  and  it  furnishes 
a  basis  for  a  truer  comprehension  by  those  who  do  go  on. 

Such  courses  are  deservedly  popular  in  Europe,  but  no  good  American  geometric 
notebook  exists.  This  is  modeled  after  those  long  used  successfully  in  England,  some 
of  which  have  been  extensively  used  in  America. 


INSTRUMENTS 

THE  student  should  have  the  following  instruments : 

1.  A  ruler,  of  which  one  edge  is  divided  into  inches  and  eighths  of  an  inch. 
Obtain  if  possible  a  ruler  on  which  the  metric  units,  centimeters  and  millimeters, 

are  also  marked. 

2.  A  good  pair  of  compasses,  with  pen  and  pencil  points. 

3.  A  semicircular  protractor  (see  p.  46). 


4.   A  drawing  triangle,  preferably  one  having  angles  of  90°,  60°,  30°. 

B 


5.   One  soft  and  one  medium  hard  pencil. 

Any  reasonably  good  case  of  drawing  instruments  will  contain  these  and  other 
desirable  instruments. 

In  working  out  the  problems  of  Section  12,  pages  55-56,  a  supply  of  paper  ruled 
in  squares  will  be  needed. 


CONSTRUCTIVE   GEOMETRY 


CONSTRUCTIVE    GEOMETRY 


i.   DEFINITIONS  AND   STATEMENTS  OF  FACT 

A  point  is  represented  in  a  drawing  by  a  dot  or  by  some  other  small  mark.  We 
try  to  make  the  dot  as  small  as  we  can,  but  it  must  be  large  enough  to  be  seen.  Men- 
tally, we  think  of  a  point  as  having  no  width  nor  breadth,  but  it  would  be  unreasonable 
to  expect  to  make  an  actual  dot  without  thickness  or  breadth. 

Lines  are  drawn  by  moving  the  pencil  point  on  the  paper.  As  before,  we  think 
of  a  line  mentally  as  without  width,  but  the  pencil  marks  we  make  in  a  drawing  must 
be  heavy  enough  to  be  seen. 

Lines  may  be  straight  or  curved.  A  good  idea  of  a  straight  line  is  formed  by  means 
of  a  tightly  stretched  string,  or  by  sighting  between  two  points.  A  line  may  be  tested 
for  straightness  by  trying  to  fit  the  edge  of  a  ruler  to  it.  If  the  line  is  curved,  the  ruler 
will  not  fit  it,  but  can  be  made  to  cross  it  at  least  twice. 

A  ruler  may  be  tested  for  straightness  by  sighting  along  its  edge.  Another  test 
for  straightness  of  the  ruler  is  to  draw  a  line  along  its  edge  on  paper  and  then  turn  the 
ruler  over  and  fit  the  edge  to  the  line  in  several  positions.  If  the  edge  fits  the  line  in 
all  positions,  the  line  is  straight  and  the  ruler  is  good. 

A  very  good  straight  edge  may  be  made  by  folding  a  piece  of  paper  in  the  ordinary 
manner.  Thus  the  edge  of  an  envelope  is  usually  quite  straight. 


CONSTRUCTIVE  GEOMETRY 


2.   MEASUREMENT  OF  DISTANCES 

The  scale  on  a  ruler  marked  in  inches  is  usually  subdivided  into  eighths  or  into 
sixteenths  of  an  inch.  Twelve  inches  make  a  foot.  Three  feet  make  a  yard.  What 
other  units  of  length  in  this  English  system  do  you  know  ? 

The  scale  on  a  ruler  marked  in  centimeters  is  usually  subdivided  into  tenths  of  a 
centimeter;  that  is,  into  millimeters.  Ten  millimeters  make  a  centimeter.  One 
hundred  centimeters  make  a  meter.  A  meter  is  about  forty  niches  (more  exactly, 
39.37  inches). 

The  meter  is  the  basis  of  the  so-called  metric  system.  A  table  of  units  in  that 
system  is  usually  given  in  arithmetic,  and  can  be  found  also  in  any  encyclopedia. 

Familiarity  with  these  units  of  length  is  gained  by  estimating  the  lengths  of  lines 
drawn  on  paper,  and  the  lengths  of  actual  objects,  and  by  comparing  these  estimates 
with  actual  measurements  with  a  ruler. 


EXERCISES   I 

i.   Estimate  the  length  of  each  of  the  following  lines,  then  measure  each  of  them; 
note  your  error. 

A  B 


D       E 
FIG.  3 


Enter  your  results  on  page  3  in  a  table  as  follows : 


ESTIMATED  LENGTH 

MEASURED  LENGTH 

ERROR  IN  ESTIMATE 

Line  AB 

Line  CD 

Line  EF 

In  making  such  a  table,  draw  the  lines  neatly  with  a  ruler.  The  corners  can  be 
made  quite  true  by  using  the  square  corner  of  your  drawing  triangle. 

One  purpose  in  this  course  is  to  learn  to  draw  such  lines  as  those  in  this  table 
neatly  and  accurately.  To  this  end,  use  your  compasses  as  well  as  your  ruler,  to  make 
columns  of  equal  width. 


CONSTRUCTIVE  GEOMETRY  5 

2.  Measure  the  lengths  of  AB,  BC,  and  AC  in  the  following  figure,  separately. 
Add  the  measured  lengths  of  AB  and  of  BC,  and  see  how  nearly  the  sum  comes  to  AC. 
Enter  your  results  in  a  neatly  drawn  table. 


FIG.  4 


3.  Estimate  the  length  of  this  page ;  its  width;  the  length  of  the  cover ;  the  thick- 
ness of  the  book.     Measure  these  same  distances,  and  note  the  errors  in  your  estimates. 
Enter  all  these  numbers  in  a  neatly  drawn  table. 

4.  Draw  a  straight  line,  and  mark  on  it  two  points  A  and  B  which  are  i|  inches 
apart.     Then  mark  a  third  point  C  so  that  BC  =  i^  inches.     Measure  AC.     Compare 
the  length  of  AB  +  the  length  of  BC  with  the  length  of  AC. 

5.  Draw  a  straight  line  and  mark  on  it  four  points  A,  B,  C,  D,  in  that  order,  and  so 
that  AB  =  i|  in.,  BC  =  £  in.,  CD  =  i|  in.     Measure  AD  and  compare  it  with  AB 
+  BC  +  CD. 

6.  Draw  a  straight  line,  and  mark  on  it  five  points  A ,  B,  C,  D,  E,  one  inch  apart. 
Measure  the  lengths  AB,  AC,  AD,  AE,  on  your  centimeter  scale.     Enter  the  results  in 
a  table  similar  to  the  following  one. 


INCHES 

CENTIMETERS 

CENTIMETERS  IN  1  INCH 

AB 

I 

AC 

2 

AD 

3 

AE 

4 

The  last  column  is  to  be  filled  out  by  dividing  the  number  of  centimeters  in  each 
of  the  lengths  measured  by  the  number  of  inches  in  it.     Do  your  results  agree  precisely  ? 

7.  Draw  a  straight  line  and  mark  several  points  on  it  one  centimeter  apart.     Meas- 
ure the  distances  from  the  first  point  to  each  of  the  others  in  inches.     Make  a  table 
similar  to  that  of  Ex.  6,  with  the  headings  in  the  order :  centimeters,  inches,  inches  in  I 
centimeter. 

8.  Measure  the  width  and  length  of  your  desk  in  feet  and  inches;    reduce  the 
results  to  inches.     Measure  the  same  distances  in  centimeters.     Make  a  table  similar 
to  that  of  Ex.  6  for  these  measurements. 


6  CONSTRUCTIVE  GEOMETRY 

3.   DIVISION  OF  A  LENGTH  INTO  EQUAL  PARTS 

It  is  often  convenient,  for  example  in  making  such  tables  as  that  of  Ex.  i,  p.  2, 
to  divide  a  length  into  two  or  more  equal  parts. 
This  can  be  done  in  several  ways : 

(a)  Arithmetically,  by  measuring  the  given  length,  dividing  the  result  into  the 
desired  number  of  parts,  and  then  marking  points  at  distances  from  each  other  equal 
to  the  quotient. 

(b)  Mechanically,  by  paper  folding,  or  by  some  similar  scheme.    A  sheet  of  paper 
may  be  folded  very  easily  into  two,  four,  eight,  etc.,  equal  strips. 

(c)  Geometrically,  without  first  measuring  the  line.     Later  we  shall  see  how  to  do 
this  directly.     Just  now  it  can  be  done  by  trial  by  means  of  compasses.     A  very  few 
trials  will  give  a  good  result. 

EXERCISES   II 

1.  Draw  a  straight  line  and  mark  two  points  on  it.     Divide  the  length  between 
the  two  points  into  two  equal  parts  by  each  of  the  methods  just  mentioned.     Which 
method  seems  most  accurate? 

2.  Mark  two  points  on  a  straight  line  as  above.     Divide  that  part  of  the  line 
between  the  two  points  into  three  equal  parts.     Which  of  the  methods  described  above 
can  be  used  conveniently? 

3.  Divide  the  part  of  a  line  between  two  points  on  it  into  four  equal  parts.     Which 
methods  are  convenient  ? 

4.  Draw  a  triangle  of  any  shape.     Measure  each  of  its  sides. 

Divide  one  of  the  sides  into  two  equal  parts.  Draw  a  line  from  the  opposite  corner 
of  the  triangle  to  the  middle  point  of  the  side.  Such  a  line  in  a  triangle  is  called  a 
median.  Measure  the  length  of  this  line. 

5.  Draw  a  triangle  of  any  shape.     Divide  each  of  the  sides  into  two  equal  parts. 
Draw  all  the  possible  medians. 

How  many  medians  are  there  ?  A  test  for  the  correctness  of  the  drawing  is  that 
all  the  medians  should  pass  through  a  common  point. 

6.  Draw  a  triangle  of  any  shape  and  divide  each  of  its  sides  into  two  equal  parts. 
Join  the  middle  points  of  two  of  the  sides  by  a  straight  line.     Measure  the  length  of  this 
line.    How  does  its  length  compare  with  the  length  of  the  third  side  of  the  triangle  ? 

7.  Draw,  in  Ex.  6,  the  other  two  lines  which  connect  the  middle  point  of  one  side 
with  the  middle  point  of  another.     Shade  the  interior  of  the  small  triangle  formed  by 
such  lines. 

Can  you  convince  yourself  that  the  original  triangle  is  now  divided  into  four  small 
triangles,  and  that  the  sides  of  each  of  them  are  exactly  half  the  length  of  the  sides  of 
the  original  one? 


CONSTRUCTIVE  GEOMETRY 


II 


4.  TO  DRAW  CIRCLES 

The  compasses  are  useful  for  measuring  distances.  They  may  be  used  for  laying 
off  on  a  line  a  distance  equal  to  that  between  two  points  on  another  line. 

Circles  are  usually  drawn  by  means  of  compasses. 

The  point  at  which  the  fixed  point  of  the  compasses  is  placed  is  called  the  center  of 
the  circle.  The  line  traced  by  the  other  (moving)  point  of  the  compasses  is  called  the 
circumference  of  the  circle,  or  simply  the  circle. 

The  distance  from  the  center  to  the  circumference  is  called  the  radius  of  the  circle. 


EXERCISES  in 

1.  Open  the  compasses  so  that  the  distance  between  the  two  points  is  i  inch. 
Draw  a  circle,  keeping  this  opening  fixed. 

2.  (a)  Draw  a  straight  line.     About  some  point  on  this  line  draw  a  circle.     In  how 
many  points  does  the  circle  cut  the  straight  line  ? 


FIG.  5 


(6)  From  one  of  the  points  where  the  circumference  cuts  the  line  draw  a  circle  with 
the  same  radius  as  the  original  circle. 

(c)  Draw  a  line  connecting  the  points  where  the  two  circles  meet  each  other. 

3.  Draw  two  circles,  each  2\  in.  in  radius,  about  two  points  4  in.  apart  on  a  line. 
Connect  the  two  points  in  which  these  circles  meet  each  other  by  a  straight  line. 

4.  Draw  two  equal  circles,  each  i  in.  in  radius  about  points  2  in.  apart  on  a  line. 
In  how  many  points  do  these  circles  meet  each  other  ? 

How  far  apart  are  the  centers  of  two  circles,  if  the  circles  just  touch  each  other  in 
one  point? 


12  CONSTRUCTIVE  GEOMETRY 

5.  (a)  Draw  any  two  circles  which  cut  each  other  in  two  points,  and  draw  the  line 
joining  their  centers. 

(6)  Draw  the  line  joining  the  two  points  where  they  cut  each  other. 

These  two  lines  are  perpendicular  to  each  other ;   that  is,  they  come  together  at  a 
square  corner,  which  will  fit  the  square  corner  of  the  drawing  triangle. 

6.  (a)  Draw  two  equal  circles  so  that  the  circumference  of  one  passes  through  the 
center  of  the  other,  and  draw  a  straight  line  joining  their  centers. 

(b)  Join  both  centers  to  one  of  the  points  in  which  the  circles  cut  each  other. 


FIG.  6 


The  three  lines  form  a  triangle,  all  three  of  whose  sides  are  equal.  It  is  called  an 
equilateral  triangle. 

7.  In  Fig.  5,  OC  =  CO'  and  BC  =  AC. 

[This  is  true  because  the  two  circles  are  the  same  size.  Hence  we  can  pick  up  the  whole  figure, 
turn  it  over,  and  lay  it  down  again  with  O'  where  O  was  and  O  where  O'  was.  Likewise,  the  figure  can 
be  turned  so  that  B  falls  where  A  was,  and  A  where  B  was.] 

Draw  Fig.  5  again,  and  draw  a  circle  about  C  as  center  with  a  radius  CO.  Does  it 
pass  through  O'  ? 

Draw  a  circle  about  C  as  center  with  a  radius  CA .    Does  it  pass  through  B  ? 

8.  Draw  an  equilateral  triangle  (Ex.  6)  and  draw  its  medians  (Ex.  4,  p.  6).     Notice 
that  the  median  from  A  through  the  middle  point  of  OO'  should  pass  through  B. 

9.  Redraw  the  figure  for  Ex.  6  (Fig.  6),  but  omit  OO'  and  draw  OB  and  BO'.    The 
resulting  four-sided  figure  AOBO'  has  all  four  sides  equal.    It  is  called  a  rhombus. 

[The  student  should  also  try  to  see  what  figures  are  formed  when  the  circles  in  Exs.  6,  7,  8,  9,  are 
of  unequal  size,  and  when  the  center  of  one  is  not  on  the  other.] 


CONSTRUCTIVE  GEOMETRY  15 

5.  PERPENDICULARS 

EXERCISES  IV 

To  divide  a  line  into  two  equal  parts  without  measuring  it,  we  may  proceed  almost  as 
in  Ex.  5,  p.  1 2,  by  the  following  method : 

i.    (a)  Draw  a  line  and  mark  any  two  points  on  it ;   call  them  A  and  B. 
(b)  About  A  as  center  draw  a  circle  which  reaches  nearly  to  B.    About  B  as  center 
draw  a  circle  equal  to  the  first  one. 

XL, 


(c)  Connect  the  two  points  C  and  D  in  which  these  two  circles  cut  each  other  by 
a  new  straight  line.  Mark  the  point  E  where  this  new  straight  line  CD  cuts  the  line  AB. 

This  point  E  is  halfway  between  A  and  B ;  that  is,  AE  =  EB. 

The  reasons  for  this  are  exactly  similar  to  those  given  in  Ex.  7,  p.  12. 

NOTE.  After  some  practice,  the  student  will  see  that  it  is  not  necessary  to  draw 
the  full  circles,  but  only  portions  of  them,  as  in  the  printed  figure. 

2.  Draw  a  figure  which  shows  how  to  divide  a  line  joining  two  points  into  four 
equal  parts  without  measuring  it. 

3.  The  line  CD  of  Ex.  i  is  perpendicular  to  AB  at  E  (see  Ex.  5,  p.  12). 

To  draw  a  line  perpendicular  to  a  given  line,  at  a  given  point  on  that  line,  we  may 
proceed  as  follows : 

Draw  a  line  AB  and  mark  a  point  C  on  it.     On  opposite  sides  of  C  mark  two 


\Q 


c 

FIG.  8 


points  P  and  Q,  so  that  PC  =  CQ.    This  can  be  done  with  the  compasses. 

Now  follow  the  directions  of  Ex.  i  to  get  a  new  line  CD ;   this  new  line  is  perpen- 
dicular to  AB  at  C. 


1 6  CONSTRUCTIVE  GEOMETRY 

To  draw  a  line  perpendicular  to  a  given  line,  through  any  point  on  the  paper,  we  may 
proceed  as  follows : 

4.   (a)  Draw  a  line  AB  and  mark  a  point  P  not  on  the  line. 


S/ 


Vr 
/  s? 

FIG.  9 

(b)  About  P  as  center  draw  a  circle  which  cuts  the  line  in  two  points  R  and  S. 

(c)  Now  follow  the  directions  of  Ex.  i  to  find  a  new  line  CT  perpendicular  to  AB. 
This  new  line  passes  through  P ;  it  is  the  line  desired. 

5.  Draw  a  straight  line  and  mark  two  points  A  and  B  one  inch  apart  on  it.    At 
A  and  at  B  draw  lines  perpendicular  to  AB. 

Mark  a  point  C  on  the  perpendicular  through  A  and  one  inch  above  A.  Mark  a 
point  D  on  the  perpendicular  through  B  and  one  inch  above  B. 

Connect  C  and  D  by  a  straight  line.    The  figure  A  BCD  is  a  square. 

6.  Carry  out  the  same  directions  as  in  Ex.  12,  except  that  AC  and  BD  are  each 
one  inch  long,  while  AB  is  of  different  length.     Such  a  figure  is  a  rectangle. 

In  a  square,  each  side  is  perpendicular  to  the  sides  next  to  it,  and  all  the  sides  are  of  equal  length. 

In  a  rectangle,  each  side  is  perpendicular  to  the  sides  next  to  it,  and  each  side  is  equal  to  the  side  op- 
posite it. 

In  a  rhombus  (Ex.  9,  p.  12)  all  four  sides  are  of  equal  length,  but  the  sides  meet  at  any  angle  we  may 
wish. 

7.  Draw  a  rectangle  four  inches  long  and  \  inch  high.     Divide  this  rectangle  into 
two  equal  rectangles  by  means  of  a  perpendicular  at  the  middle  point  of  the  base. 
Divide  these  rectangles  again  into  two  equal  parts. 

In  this  way  very  accurate  blank  forms,  such  as  that  used  on  p.  2,  may  be  made. 

8.  A  man  goes  2  miles  east,  then  8  miles  north,  and  finally  4  miles  west.     Draw 
a  map  of  his  route.     Measure  the  distance  from  his  starting  point  to  his  final  position. 

9.  Draw  a  small  map  of  the  city  block  on  which  your  school  stands,  and  of  each  of 
the  blocks  next  to  it.    Allow  for  widths  of  streets. 

Measure  the  distance  between  two  corners  not  on  the  same  street. 


CONSTRUCTIVE  GEOMETRY 


21 


6.   PARALLELS 

Two  lines  perpendicular  to  the  same  line  will  never  meet  each  other. 

are  called  parallels. 

EXERCISES  v 


Such  lines 


1.  Draw  a  line  and  mark  several  points  A,  B,C,  —  on  it.    Draw  perpendiculars 
at  each  of  the  points  A ,  B,  C,  •••  to  the  line.    These  new  lines  are  all  parallel. 

2.  To  draw  a  parallel  to  a  given  line  through  a  given  point: 
(a)  Draw  a  line  AB  and  mark  a  point  P  not  on  the  line. 


given 


FIG.  10 

(b)  Draw  a  second  line  through  P  perpendicular  to  the  given  line  and  draw  a 
third  line  through  P  perpendicular  to  the  second  line.     (See  Ex.  3,  p.  15.) 

The  third  line  is  parallel  to  the  first,  since  both  are  perpendicular  to  the  second  line. 

3.    To  draw  parallels  with  the  drawing  triangle. 

(a)  Draw  a  line  AB,  and  mark  a  point  P,  not  on  it. 


(6)  Lay  the  drawing  triangle  with  any  edge  fitting  the  given  line.    Place  a  ruler 
(or  a  book)  so  as  to  fit  either  of  the  other  edges  of  the  drawing  triangle. 

(c)  Hold  the  ruler  (or  the  book)  still,  and  slide  the  triangle  along  it,  keeping  the 
edge  of  the  triangle  tightly  fitted  against  the  ruler  (or  book)  until  the  edge  of  the 
triangle  which  did  fit  against  the  given  line  comes  near  P. 

(d)  Draw  a  line  through  P  along  that  side  of  the  triangle  which  did  fit  the  given  line. 
The  new  line  is  parallel  to  the  given  line. 

4.  Draw  a  picture  of  a  picket  fence  by  drawing  two  very  long  rectangles  to  repre- 
sent the  horizontal  rails,  and  smaller  rectangles  to  represent  the  slats. 

5.  Draw  an  ornamental  border  by  drawing  four  rectangles  one  inside  another 
about  |  in.  apart.    This  may  be  decorated  by  shading. 


22 


CONSTRUCTIVE  GEOMETRY 


6.  Draw  nine  parallels  |  in.  apart,  and  nine  parallels  perpendicular  to  them  \  in. 
apart. 

Shade  the  alternate  squares  to  represent  a  checkerboard. 

An  ornamental  border  may  be  placed  around  the  whole  figure,  as  in  Ex.  5. 

7.  Mark  several  points  A,  B,  C,  D,  etc.,  on  your  paper.     Draw  lines  through  each 
of  them  parallel  to  the  top  edge  of  the  paper  by  means  of  Ex.  3. 

Likewise  draw  lines  parallel  to  one  side  edge  of  the  paper  through  A,  B,  C,  D. 

8.  To  draw  perpendiculars  with  the  drawing  triangle. 

The  right-angled  corner  of  the  drawing  triangle  may  be  used  directly,  as  in  Ex.  i, 
p.  2.    This  gives  blunt  corners  which  are  unsightly. 

A  better  result  is  obtained  by  using  the  triangle  as  in  the  accompanying  figure. 
The  principle  is  almost  the  same  as  in  Ex.  3,  but  the  triangle  and  the  ruler  are  placed 


as  shown  in  figure :  the  number  i  marks  the  first  position  of  the  triangle,  fitting  against 
the  given  line  AB ;  the  number  2  marks  the  position  of  the  ruler,  fitting  against  the 
triangle ;  the  number  3  marks  the  second  position  of  the  triangle,  after  it  has  slid 
along  the  ruler  to  the  given  point  P.  Draw  such  a  figure. 

9.  Mark  several  points  A,  B,  C,  D,  on  your  paper.     Draw  lines  perpendicular  to 
the  top  edge  of  your  paper  through  each  of  these  points  by  means  of  Ex.  8. 

10.  Mark  a  point  P  on  your  paper.     Draw  a  line  through  P  parallel  to  the  top  edge 
of  the  paper.     Draw  another  line  through  P  perpendicular  to  one  of  the  side  edges  of 
your  paper.     If  the  paper  is  cut  true,  and  if  you  have  drawn  accurately,  these  two 
lines  should  be  exactly  the  same. 

11.  Draw  a  map  showing  at  least  four  or  five  principal  streets  running  east  and 
west,  and  an  equal  number  running  north  and  south,  in  the  city  in  which  you  live. 
Use  Exs.  3  and  8.     Measure  the  distance  on  this  map  between  two  important  points 
not  on  the  same  street. 


CONSTRUCTIVE  GEOMETRY  25 


7.  DRAWING  ORNAMENTAL  PATTERNS 

Many  ornamental  designs  may  be  made  by  means  of  the  previous  constructions. 
Some  of  these  follow.    Let  the  student  try  to  devise  others. 


EXERCISES   VI 

1.  Draw  a  rectangle  3  inches  high  and  i|  inches  wide.     Draw  a  half  circle 
whose  center  is  the  middle  point  of  the  top  side  of  the  rectangle  and  whose  radius 
is  f  inch. 

This  is  the  form  of  the  so  called  Roman  window,  surmounted  by  a  circular 
arch.  Ornament  it  by  lines  drawn  about  £  inch  from  each  side  and  by  another 
half  circle  with  the  same  center  and  with  a  radius  about  £  inch  larger  than  that  of  the 
first  circle. 

Other  lines  may  be  drawn  hi  an  ornamental  pattern  to  represent  frames  of  glass. 

2.  (a)  Draw  an  equilateral  triangle  as  in  Ex.  6,  p.  12. 

(b)  About  each  corner  of  the  base  as  center  draw  a  portion  of  a  circle  joining  the  two 
remaining  corners. 

This  is  the  basis  of  the  so-called  Gothic  window.    Compare  Ex.  13,  p.  65. 

3.  Draw  a  square.    About  each  of  the  corners  as  a  center  draw  a  circle  whose 
radius  is  equal  to  one  side  of  the  square. 

Various  ornamental  patterns  may  be  formed  by  drawing  only  a  part  of  each  circle, 
and  by  shading  the  figures.  Repeating  the  same  design  in  several  squares  gives  a  strik- 
ing effect. 

4.  Draw  an  equilateral  triangle.     Connect  each  corner  by  a  straight  line  to  the 
middle  point  of  the  opposite  side.      The  three  new  lines  thus  drawn  meet  hi  one 
common  point. 

About  this  common  point  as  center  draw  two  circles,  one  of  which  passes  through 
each  of  the  corners,  while  the  other  just  touches  each  side  of  the  triangle. 

5.  Draw  a  triangle  with  unequal  sides,  and  divide  it  into  four  small  triangles 
as  in  Ex.  7,  p.  6,  by  drawing  the  lines  connecting  the  middle  points  of  the  sides. 

Repeat  the  process,  so  that  each  of  the  four  small  triangles  is  divided  into  four  still 
smaller. 

By  repeating  this  process  and  then  shading  the  very  smallest  triangles  alternately, 
a  variety  of  interesting  patterns  may  be  formed. 


26 


CONSTRUCTIVE  GEOMETRY 


6.    (a)  Mark  three  points  P,  Q,  R  one  inch  apart  on  a  straight  line.    About  each 
point  as  center  draw  a  circle  of  radius  one  inch. 


(b)  Mark  the  four  points  A ,  B,  C,  D,  in  which  the  middle  circle  cuts  the  others. 
Join  the  six  points  A,  B,  R,  D,  C,  P  by  straight  lines  to  form  a  six-sided  figure. 
This  six-sided  figure  is  called  a  regular  hexagon. 

7.  Redraw  Fig.  13,  and  erase  all  except  the  central  circle  and  the  hexagon  ABPCDR. 


FIG.  14 

About  each  of  the  six  corners  of  the  hexagon  draw  a  circle  which  passes  through 
the  center  Q  of  the  original  central  circle,  marking  only  the  parts  which  lie  inside 
the  original  circle.  Shade  parts  of  the  figure  to  bring  out  the  pattern  vividly. 

8.   Draw  each  of  the  following  figures : 


FIG.  15 


CONSTRUCTIVE  GEOMETRY 


8.   MEASUREMENT  OF  ANGLES 

The  angle  between  two  perpendicular  lines  is  called  a  right  angle.  It  is  divided  into 
90  equal  parts,  each  of  which  is  called  a  degree  (°).  One  sixtieth  of  a  degree  is  called 
a  minute  (') ;  one  sixtieth  of  a  minute  is  called  a  second  (")• 

Since  there  are  four  complete  right  angles  formed  at  the  point  where,  two  perpen- 
diculars meet,  the  total  angle  around  the  point  is  4  X  90  degrees,  or  360  degrees. 


EXERCISES  vn 

1.  Draw  a  square.    How  many  right  angles  does  it  have  ?    What  is  the  sum  of  all 
of  them  put  together  ? 

2.  Draw  a  square.     Connect  the  opposite  corners  by  straight  lines. 

These  lines  are  called  diagonals.    The  diagonals  divide  each  of  the  angles  at  the 
corners  into  two  equal  parts.     How  large  is  each  of  these  parts  ? 

3.  Draw  an  equilateral  triangle.     The  size  of  each  angle  is  60°.  What  is  the  sum 
of  all  three  of  them  ? 

4.  Draw  an  equilateral  triangle  ABC,  on  the  base  AB.     Extend  the  side  AB 
beyond  B.    At  B  draw  a  perpendicular  BD  to  AB,  above  AB. 

How  large  is  the  angle  ABC?    How  large  is  angle  ABD?    CBD? 

5.  To  move  an  angle  from  one  position  to  another. 

(a)  Draw  an  angle  ABC,  with  its  corner  at  B. 

(b)  Draw  any  new  line  M N  and  mark  a  point  P  on  it. 


FIG.  i 8     - 

(c)  About  A  as  center  draw  a  circle  of  convenient  size  which  cuts  AB  at  B  and 
AC  at  C. 

(d)  About  P  as  center  draw  a  circle  of  the  same  size  as  the  preceding  one,  and 
mark  a  point  Q  where  this  circle  cuts  the  line  through  P. 

(e)  With  a   radius   equal   to  BC  draw  a  circle  about  Q  as  center,  and  let  0  be 
a  point  where  this  circle  cuts  the  one  whose  center  is  P. 

The  angle  QPO  is  the  same  as  the  angle  ABC  moved  into  a  new  position. 


32 


CONSTRUCTIVE  GEOMETRY 


5.  To  draw  a  clockface,  first  draw  a  circle  and  mark  its  center. 

Then  make  successive  angle  of  30°  (Ex.  4)  whose  corners 
are  at  the  center  of  a  circle,  beginning  with  a  vertical  line 
through  the  center  of  the  circle. 

Mark  the  points  along  the  circumference  XII,  I,  II,  III, 
IIII,  V,  VI,  etc.,  as  on  a  clockface. 

This  may  be  further  ornamented  as  in  the  figure. 

6.   Draw  an  angle  of  45°  (Ex.  2),  and  another  angle  of 
30°  (Ex.  4).    Move  the  second  angle  to  a  new  position  so 
that  its  corner  is  at  the  corner  of  the  45°  angle  and  so  that 
one  side  of  each  lies  in  the  same  line. 
What  is  the  difference  between  the  two  angles?     Shade  the  corresponding  angle 
in  your  figure. 

7.    To  divide  any  angle  into  two  equal  parts. 
(a)  Draw  an  angle  ABC,  with  the  corner  at  B. 

(V)  With  B  as  center,  and  with  any  radius  you  please,  draw  a  circle  to  cut  AB 
at  some  point  M  and  to  cut  CB  at  some  point  L. 


FIG.  19 


(c)  With  L  as  center  and  with  any  radius  you  please,  draw  a  circle. 

(d)  With  M  as  center  draw  a  circle  of  the  same  radius  as  that  about  L. 

(e)  Mark  the  point  G  where  the  last  two  circles  cut  each  other. 
(/)  Draw  the  straight  line  BG. 

Then  BG  divides/  ABC  into  two  equal  parts,  so  that  Z  CBG  =  Z  GBA. 

8.  Draw  a  right  angle.     Divide  it  into  two  equal  parts.     How  many  degrees  are 
there  in  each  half  ? 

9.  Draw  an  angle  of  22^  degrees. 

10.  Draw  an  equilateral  triangle  (Ex.  6,  p.  12).     Divide  one  of  its  angles  into  two 
equal  parts.    How  large  is  each  half  ?     (See  Exs.  3,  4,  p.  31.) 

11.  Draw  an  angle  of  15°. 

12.  Draw  any  triangle. 

Divide  each  of  its  angles  into  two  equal  parts. 

The  three  dividing  lines  should  pass  through  a  single  common  point. 


CONSTRUCTIVE  GEOMETRY  35 


9.  TRIANGLES 

EXERCISES   WE 

i .   To  copy  a  triangle  by  means  of  Us  sides  alone. 

(a)  Draw  any  triangle  ABC. 

(b)  Draw  any  line  /.    On  /  mark  any  point  P,  and  lay  off  on  I  a  distance  PQ  equal 
to  AB  with  the  compasses. 

(c)  About  P  as  a  center,  draw  a  circle  with  a  radius  equal  to  AC ;  and  about  Q 
as  center,  draw  a  circle  with  radius  equal  to  BC. 


(d)  Mark  a  point  R  where  the  two  circles  meet,  above  /.     Draw  the  lines  PR,  QR. 
Then  the  triangle  PQR  is  exactly  the  same  size  and  shape  as  ABC. 

2.  Draw  a  triangle  whose  sides  are  respectively  3  inches,  4  inches,  2  inches,  long. 

3.  Choose  any  three  trees  in  your  school  grounds,  or  in  some  park.     Measure  the 
distances  between  each  pair,  in  feet.     Draw  a  diagram  on  paper  to  represent  their 
positions,  using  f  inch  in  the  figure  to  every  foot  of  actual  distance. 

By  going  from  these  three  trees  to  a  fourth  one,  and  so  on,  a  diagram  may  be  made 
to  show  all  the  trees  in  a  given  yard.     This  process  is  called  triangulation. 

4.  To  copy  a  triangle  by  means  of  one  angle  and  two  sides. 

(a)  Draw  a  triangle  ABC. 

(b)  Move  the  angle  at  B  to  any  desired  new  position,  by  means  of  Ex.  5,  p.  31. 


B 
FIG.  22 

(c)  On  the  two  sides  of  this  new  angle,  lay  off  lengths  equal  to  BA  and  BC, 
respectively ;   and  join  the  ends  of  these  lengths. 

The  new  triangle  formed  is  precisely  the  same  size  and  shape  as  the  triangle  ABC. 

5.  Draw  a  triangle  of  which  one  side  is  2  inches  long,  another  side  3  inches  long,  and 
the  angle  between  them  is  60°.     Measure  the  third  side  with  your  ruler.     How  long  is  it  ? 

6.  Draw  a  triangle  with  two  sides  i  inch  and  3  inches  long,  respectively,  and  with 
the  angle  between  them  equal  to  90°.     Measure  the  third  side. 


CONSTRUCTIVE  GEOMETRY 

7.   To  copy  a  triangle  by  means  of  one  side  and  two  angles. 

(a)  Draw  any  triangle  ABC. 

(b)  On  any  desired  line  lay  off  a  length  PQ  equal  to  AB. 


(c)  At  P  make  an  angle  equal  to  Z  BA C,  and  at  Q  an  angle  equal  to  Z.  ABC. 

(d)  Extend  the  sides  of  these  new  angles  to  form  a  triangle  PQR. 
This  new  triangle  is  precisely  the  same  shape  and  size  as  ABC. 

8.  Draw  a  triangle  of  which  one  side  is  2  inches  long,  with  an  angle  of  45°  at  one 
end  of  that  side  and  an  angle  of  60°  at  the  other  end. 

Measure  the  two  sides  which  were  not  given. 
Measure  the  third  angle  of  the  triangle. 

9.  From  a  line  100  ft.  long,  near  the  shore  of  a  lake, 
a  surveyor  measures  the  angles  to  an  island  in  the  water. 
If  these  angles  are  90°  and  30°  respectively,  how  far  is 
the  island  from  the  shore  line  ? 

Draw  a  figure  with  i  inch  in  the  figure  equal  to  25  feet  of  actual  distance. 

10.  A  flagpole  FT  stands  on  a  level  field.     From  a  point  S,  which  is  80  feet  away 
from  F,  it  is  found  that  the  angle  FST  is  equal  to  60°. 

Draw  a  figure,  making  i  inch  in  your  figure  equal  to  20  feet 
of  actual  distance.     Measure  FT.    How  high  is  the  flagpole  ? 

11.  From  the  window  of  a  lighthouse  known  to  be  75  feet 
above  the  water  level,  a  boat  is  seen,  at  an  angle  of  15°  below  the 
horizontal  line  through  the  window.     How  far  away  is  the  boat  ? 

12.  A  man  walks  3  miles,  turns  45°  to  his  right,  goes  2  miles,  turns  90°  to  his 
right,  and  goes  i  mile.    How  far  is  he  from  his  starting  point  ? 

13.  How  long  a  ladder  is  needed  to  reach  the  top  of  a  wall  20  feet  high,  if  the  foot 
of  the  ladder  must  be  10  feet  from  the  side  of  the  wall? 

14.  Suppose  a  surveyor's  notes  of  a  triangular  field  read :    Base  AB  =  100  feet, 
angle  at  A ,  60°,  angle  at  B,  90°.     Draw  a  plan  of  the  field,  letting  i  inch  in  your  plan 
equal  50  feet  of  actual  distance,  and  measure  the  two  other  sides. 


CONSTRUCTIVE  GEOMETRY  41 

15.  Suppose  the  notes  of  a  triangular  field  to  be  AB  =  60  yards,  AC  =  45  yards, 
angle  at  A  =  60°.     Draw  a  plan  of  the  field,  and  find  the  length  of  BC. 

1 6.  A  man  walks  3  miles  and  turns  30°  to  his  right.    He  then  walks  4  miles,  and 
turns  60°  to  his  right,  and  again  walks  3  miles.     Find  how  far  he  is  from  his  starting 
point. 

17.  A  and  B  are  two  forts  separated  by  a  river.    A  man  goes  to  a  bridge,  C,  from 
one  of  the  forts,  and  starts  back  to  the  other  fort  on  a  straight  road  making  an  angle 
of  30°  with  the  road  on  the  other  side  of  the  river.     It  is  6  miles  from  A  to  the  bridge, 
and  8  miles  from  B  to  the  bridge.    How  far  apart  are  the  forts  ? 

18.  A  man  100  ft.  from  the  base  of  a  wireless  station  tower  finds  that  the  angle 
to  the  top  of  the  tower  is  60°.     Draw  a  plan,  and  measure  the  height  of  the  tower. 

19.  A  baseball  diamond  is  a  square  90  ft.  on  each  side,  the  bases  being  at  the 
corners.     (See  Fig.  26.)     If  a  ball  is  caught  halfway  between  second  base  and  third 
base,  find  the  distance  to  first  base ;  find  the  distance  to  home  plate. 


FIG.  26 

20.  An  upright  pole,  30  feet  high,  is  stayed  by  a  rope  carried  from  the  top  to  a 
point  on  the  ground  20  feet  from  the  foot  of  the  pole.     Make  a  diagram  of  this,  using 
i  inch  =  io  ft.,  and  find  the  length  of  the  rope. 

21.  Directly  east  of  where  a  man  stands  he  can  see  a  church  tower  which  he 
knows  to  be  two  miles  distant ;  due  north  he  sees  a  standpipe  which  is  if  miles  dis- 
tant.    Draw  a  plan,  and  find  the  distance  from  the  church  to  the  standpipe. 

22.  An  automobile  runs  25  miles  north  along  a  straight  road,  and  then  runs  17 
miles  due  west.     Draw  a  plan,  and  find  how  far  the  machine  is  from  the  starting 
point.     How  many  miles  would  an  aeroplane  save,  if  it  flew  straight  across  ? 

23.  In  rowing  across  a  river  78  yards  wide,  a  man  was  carried  downstream  23 
yards.     Represent  this  on  a  plan,  and  find  the  distance  between  the  starting  point 
and  the  landing  point. 


42  CONSTRUCTIVE  GEOMETRY 

10.   DIVISIONS  OF  A  LINE.     SIMILAR  FIGURES 

EXERCISES    IX 

i.   To  divide  a  line  into  three  equal  parts. 

(a)  Draw  a  line  and  mark  two  points  A  and  B  on  it. 

(b)  From  A  draw  any  other  line  AC  so  that  Z.  BAG  is  of  any  convenient  size. 

(c)  On  AC  mark  three  points,  P,  Q,  R,  so  that  AP  =  PQ  =  QR. 

(d)  Draw  the  straight  line  BR. 

(e)  Draw  parallels  to  BR  through  P  and  Q.     (See  Ex.  3,  p.  21.) 

These  parallel  lines  divide  AB  into  three  equal  parts. 

Similarly  a  line  AD  may  be  divided  into  any  desired  number  of  equal  parts  by 
drawing  a  set  of  parallels  from  points  H,  7,  7,  K,  •••,  equally  spaced  on  some  other  line. 


2.  Draw  a  line  AB  and  divide  it  into  five  equal  parts. 

3.  Draw  a  square.     Divide  this  square  into  three  equal  rectangles,  by  lines  par- 
allel to  its  base,  through  points  that  divide  one  side  into  three  equal  parts. 

4.  (a)  Draw  a  triangle  of  any  shape. 

(6)  Divide  each  of  its  sides  into  three  equal  parts. 

(c)  Draw  a  new  triangle,  each  of  whose  sides  is  equal  to  one  third  the  correspond- 
ing side  of  the  first  triangle. 

5.  To  reduce  a  figure  in  the  ratio  i  :  3  means  to  make  a  new  figure  in  which  each 
line  is  one  third  the  corresponding  line  in  the  given  figure. 

(a)  Draw  a  rectangle  and  one  diagonal  of  it. 

(b)  Reduce  this  figure  in  the  ratio  1:3. 

Figures  are  said  to  be  similar  to  each  other  if  one  of  them  is  the  same  as  the  other 
except  that  it  is  reduced  or  enlarged  in  size. 

6.  Divide  a  line  4  in.  long  into  5  equal  parts ;  a  line  7  in.  long  into  n  equal  parts. 

7.  Draw  a  line  3  in.  long,  and  divide  it  into  8  equal  parts.    Test  afterwards  by 
measurement. 

8.  Use  the  same  method  to  bisect  a  line  of  any  convenient  length.    Then  bisect 
the  line  by  the  use  of  the  compass,  and  see  if  the  two  results  agree. 


CONSTRUCTIVE  GEOMETRY  45 

Drawings  which  represent  large  objects  are  always  made  on  a  reduced  scale.  The 
drawing  is  made  similar  to  the  object  represented  by  reducing  all  dimensions  in  the  same 
ratio.  Often  one  inch  in  the  drawing  represents  one  foot  on  the  object  represented. 

9.  Make  a  drawing  to  represent  a  four-sided  figure  which  has  two  sides  parallel 
and  one  foot  apart,  the  other  two  sides  equal,  but  not  parallel,  and  two  feet  long. 

10.  Draw  a  vertical  cross  section  of  a  ditch  which  is  4  feet  wide  at  the  top,  2  feet 
wide  at  bottom,  and  3  feet  deep.     Measure  the  length  of  the  side. 

n.  Draw  a  square.  On  each  side  of  this  square  draw  an  equilateral  triangle. 
Join  the  vertices  of  these  triangles,  and  show  by  measurement  that  the  figure  so  formed 
is  a  square.  What  is  the  ratio  of  the  side  of  this  square  to  that  of  the  original  one  ? 

12.  A  man  measures  a  four-sided  field.    He  finds  that  the  diagonals  bisect  one 
another,  and  form  an  angle  of  30°  with  each  other.    They  are  60  yards  and  80  yards, 
respectively.     Find  the  length  of  each  side  of  the  field. 

13.  A  ditch  around  a  prison  runs  close  up  to  the  prison  wall.     A  man  finds  that 
when  he  is  80  feet  away  from  the  outer  edge  of  the  ditch,  the  angle  to  the  top  of  the 
prison  wall  is  45°,  while  at  the  edge  of  the  ditch  it  subtends  an  angle  of  60°.     Find  the 
width  of  the  ditch. 

14.  House  plans  are  usually  drawn  on  such  a  scale  that  \  inch  in  the  drawing  repre- 
sents one  foot  in  the  actual  house.     Such  drawings  are  rather  large,  however. 

Draw  a  map  of  the  first  floor  of  your  home,  showing  all  windows,  doors,  and  parti- 
tion walls,  on  a  scale  of  f  inch  to  one  foot. 

15.  Any  given  length  may  be  multiplied  by  any  given  number  geometrically. 
Thus,  given  any  definite  length  MN,  let  us  multiply  it  by  3^. 

Draw  any  two  lines  AB  and  AC  meeting  at  A  at  any  convenient  angle  (say  between 
30°  and  45°). 

On  AB  mark  points  D  and  E  so  that  AD  =  i  inch  and  DE  =  35  inches. 
On  AC  mark  a  point  F  so  that  AF  =  MN. 

Connect  D  and  F.  Through  E  draw  a  line  parallel  to  DF.  This  parallel  meets 
AC  at  some  point,  say  G. 

Then  FG  =  3$  X  AF  or  3^  X  MN. 

16.  Take  a  length  MN  =  if  inches.     Multiply  it  by  2j  geometrically.     Measure 
the  resulting  line.     Multiply  if  by  2 \  by  arithmetic.     Is  the  answer  the  same  as  before  ? 


46 


CONSTRUCTIVE  GEOMETRY 


ii.  THE  PROTRACTOR 
The  protractor  may  be  used  to  lay  off  any  desired  angle,  as  well  as  to  measure  angles. 

EXERCISES  x 

i.  From  a  point  A,  30  feet  from  the  base  C,  of  a  tree,  CB,  the  angle  CAB  be- 
tween the  ground  and  the  line  from  A  to  the  top  B  of  the  tree  is  31°.  Find  the 
height  of  the  tree. 

B 


The  angle  between  a  horizontal  line,  such  as  AC,  and  a  line  such  as  AB,  from  the 
observer  A  to  a  high  object  such  as  B  is  called  the  angle  of  elevation  of  B. 

2.   It  is  often  difficult  to  reach  the  base  of  a  tree  or  other  object. 


FIG.  29 

From  a  point  A  the  angle  of  elevation  of  the  top  of  a  pine  is  25°.  From  a  point 
B,  which  is  50  feet  nearer  to  the  tree,  the  angle  of  elevation  is  55°. 

Draw  a  figure  by  first  drawing  the  line  AB  and  then  making  the  angles  BA C  and 
DEC  by  means  of  a  protractor.  Extend  all  lines  to  complete  the  figure,  and  measure  DC. 

3.  Find  the  height  of  a  statue,  if  the  angles  of  elevation  from  two  points,  one  of 
which  is  20  feet  nearer  the  statue  than  the  other,  are  35°  and  45°  respectively. 

4.  A  roof  is  built  with  a  pitch  of  one  third ;  that  is,  the  height  above  the  plate 
to  the  ridge  is  one  third  the  entire  span. 

Draw  the  figure  accurately  to  scale.  Measure  the  angle  which  the  rafters  make 
with  the  horizontal. 

Ridge 


j Span  of  Roof U 


FIG.  30 


CONSTRUCTIVE  GEOMETRY  51 

5.  The  rafters  of  a  roof  make  an  angle  of  35°  with  the  horizontal,  and  the  span  is 
32  feet.     Draw  a  figure  to  scale  and  measure  the  rise.     Find  the  pitch  of  the  roof. 

6.  Draw  two  lines  perpendicular  to  each  other  through  the  center  of  a  circle.     Mark 
the  points  where  these  lines  meet  the  circumference  of  the  circle,  and  join  these  points 
by  straight  lines. 

The  resulting  figure  is  a  square.     It  is  said  to  be  inscribed  in  the  circle. 

7.  Divide  the  total  angle  (360°)  about  the  center  of  a  circle  into  five  equal  angles, 
by  drawing  lines  which  make  angles  equal  to  one  fifth  of  360°  or  72°.     Use  the  protractor. 

Join  these  points  by  straight  lines.     The  resulting  five-sided  figure  is  called  a  regu- 
lar pentagon.     It  is  inscribed  in  the  circle. 

8.  A  regular  six-sided  figure  (hexagon)  can  be  inscribed  in  a  circle  by  drawing 
angles  of  60°  about  the  center  of  the  circle. 

Do  this  first  by  drawing  angles  of  60°  as  in  Ex.  3,  p.  31. 
Draw  the  same  figure,  using  your  protractor. 

9.  Draw  a  regular  eight-sided  figure  (octagon)  inscribed  in  a  circle. 

10.  Draw  a  regular  nine-sided  figure  inscribed  in  a  circle. 

11.  Draw  a  triangle  of  any  shape  and  measure  the  three  angles.    Add  your  answers 
together.     Is  the  sum  180°  ?    If  not,  how  much  does  it  differ  from  180°  ? 

The  true  sum  of  the  angles  of  any  triangle  is  180°. 

12.  Draw  a  triangle  with  a  right  angle  at  one  corner,  and  an  angle  of  25°  at  another. 
Measure  the  third  angle.     Is  the  sum  of  all  three  angles  180°? 

13.  An  angle  less  than  a  right  angle  is  called  an  acute  angle. 
A  triangle  with  one  right  angle  is  called  a  right  triangle. 

Draw  a  right  triangle,  and  measure  the  two  acute  angles.     What  is  their  sum? 

14.  Draw  a  right  triangle  of  which  one  acute  angle  is  36°.     How  large  is  the  other 
acute  angle  ?    Measure  it. 

15.  A  vertical  windmill  tower  50  feet  high  stands  on  level  ground.     Find  the 
angles  of  elevation  of  the  top  and  middle  point  of  the  tower  from  a  point  on  the  ground 
30  feet  away  from  the  base  of  the  tower. 

1 6.  A  flagstaff  stands  on  top  of  a  tower.    At  a  distance  of  80  feet  from  the  base  of 
the  tower,  the  angle  of  elevation  of  the  top  of  the  tower  is  found  to  be  55°,  while  the 
angle  of  elevation  of  the  top  of  the  flagstaff  is  75°.     Find  the  length  of  the  flagstaff  and 
the  height  of  the  tower. 


52  CONSTRUCTIVE  GEOMETRY 

17.  A  shore  battery  has  an  effective  range  of  4  miles.     A  ship  is  fired  upon  while 
2\  miles  NW  of  the  battery ;  she  then  turns  NE,  and  goes  2  miles.    There  she  anchors 
for  repairs,  thinking  herself  out  of  range.     Is  she  ? 

18.  From  a  point  P  I  walk  east  2  miles,  then  turn  SW  and  walk  3  miles.     I  then 
return  directly  to  P.    How  far  do  I  walk  all  together  ? 

19.  A  balloon  is  held  captive  by  a  rope  300  yards  long.     It  drifts  in  the  wind 
until  its  angle  of  elevation,  from  the  place  where  the  rope  is  tied,  is  65°.     How  high 
is  the  balloon  above  the  ground  ? 

20.  A  tower  stands  on  a  rock ;   a  man  100  yards  away  from  the  foot  of  the  rock 
finds  the  angle  of  elevation  of  the  foot  of  the  tower  to  be  25°.     When  he  is  200  yards 
away,  he  finds  the  angle  of  elevation  of  the  fop  of  the  tower  to  be  25°.     Find  the  heights 
of  the  rock  and  of  the  tower. 

21.  The  angle  of  depression  of  an  object  below  the  observer  means  the  angle 
between  a  horizontal  line  and  a  line  depressed  downwards  passing  through  the  object. 

At  the  top  of  a  mountain  it  is  found  that  the  angle  of  depression  of  a  neighbor- 
ing peak  is  5°.  If  the  difference  in  the  heights  of  the  two  mountains  is  known  to  be 
500  feet,  find  the  distance  between  the  peaks. 

22.  A  man  wishing  to  find  the  distance  of  an  enemy's  fort  measures  a  base  of  100 
yards,  and  finds  that  the  angles  at  the  ends  of  the  base  are  each  70°.    Find  how  far  the 
fort  is  from  either  end  of  the  base,  and  measure  the  third  angle. 

23.  Let  A  and  B  be  two  inaccessible  objects,  and  C  a  point  from  which  they  can 
both  be  seen.    The  angle  DCE  is  135°.     I  measure  CD  and  CE,  each  100  yards,  and 
observe  the  angles  CD  A  and  ACD,  and  find  them  to  be  30°  and  80°.     I  measure  the 
angles  BCE  and  CEB,  and  find  them  to  be  each  67^°.     Find  the  distance  between  A 
and  J3. 

24.  If  sand  is  poured  out  carefully  in  a  heap,  the  angle  which  the  side  of  the  pile 
makes  with  the  horizontal  is  always  the  same  for  the  same  grade  of  sand.     This  angle 
is  called  the  angle  of  repose. 

Measure  the  height  and  the  width  of  a  small  pile  of  sand  carefully. 
Draw  a  figure  to  represent  a  vertical  section  of  such  a  pile,  and  find  the  angle 
of  repose. 

25.  Draw  a  figure  to  scale  to  represent  a  pile  12  feet  wide  of  the  sand  used  in 
Ex.  24.    Measure  from  your  figure  the  height  of  the  pile.     Then  find  its  volume  from 
the  formula  £  7rr2/z,  where  TT  =  3^-,  r  =  radius  of  base,  h  =  height. 


CONSTRUCTIVE  GEOMETRY 


55 


i2.   SQUARED  PAPER  — AREAS 

Squared  paper  is  paper  ruled  into  little  squares.     It  may  be  bought  already  ruled. 

Usually  the  smallest  squares  are  made  one  tenth  of  an  inch  on  each  side.    Larger 
squares  whose  sides  are  one  inch  long  are  usually  marked  by  heavier  lines. 

EXERCISES    XI 

i.   Copy  the  following  designs  by  drawing  more  heavily  some  of  the  lines  on  a  sheet 
of  squared  paper,  and  by  drawing  in  the  diagonals. 


FIG.  31 

2.  Draw  a  triangle  on  squared  paper,  and  estimate  its  area. 

Remember  that  there  are  100  small  squares  whose  sides  are  ^  inch,  in  one  square 
inch.  Hence  each  small  square  counts  as  y^-  square  inch  if  the  paper  is  ruled  in 
tenths  of  an  inch. 

A  good  plan  is  to  count  all  the  squares  which  are  wholly  inside  a  figure,  and  then 
to  add  half  the  squares  which  are  partly  inside  and  partly  outside. 

3.  Draw  any  rectangle  on  squared  paper.     Draw  a  similar  rectangle  in  the  ratio 
1:2,  and  show  by  counting  the  squares  that  the  area  of  the  larger  one  is  four  times 
the  area  of  the  smaller  one. 

The  area  of  a  rectangle  in  square  inches  is  equal  to  the  number  of  inches  in  its  length 
times  the  number  of  inches  in  its  height. 

4.  Draw  any  circle.     Draw  a  circle  of  twice  the  radius.     Compare  their  areas. 

5.  Draw  a  square  and  inscribe  a  circle  in  it.     Compare  the  area  of  the  circle  with 
that  of  the  square. 

The  area  of  a  circle  is  found  to  be  about  3}  times  the  square  of  the  radius.  (More  accurately 
3.1416  X  (radius)2.) 

The  area  of  the  square  in  which  the  circle  is  inscribed  is  evidently  4  X  the  square  of  the  radius. 
The  two  areas  should  therefore  be  in  the  ratio  3^  to  4,  nearly. 


56  CONSTRUCTIVE  GEOMETRY 

6.  A  good  practical  way  to  enlarge  a  figure  is  to  draw  it  on  squared  paper  and  then 
redraw  it,  taking  as  many  tenths  of  inches  in  place  of  one  tenth  inch  as  is  desired  for  the 
enlargement. 

Draw  a  figure  of  any  kind,  and  enlarge  it  by  this  method  in  the  ratio  1:5. 

7.  Make  an  outline  map  of  the  state  of  Michigan,  twice  the  size  of  that  in  your 
atlas,  by  using  squared  paper.     Verify  the  correctness  of  your  copy  by  measuring  dis- 
tances between  other  points  than  those  used  to  make  your  figure. 

8.  Make  a  map  of  the  Mississippi  and  Ohio  rivers  from  Quincy,  111.,  and  Cincin- 
nati, to  Memphis,  half  or  twice  the  size  of  the  map  in  your  geography.    Test  the  correct- 
ness of  your  drawing. 

9.  On  a  map  whose  scale  is  5  miles  to  the  inch,  a  piece  of  land  is  represented  by  an 
area  of  24  square  inches.     What  is  the  area  of  the  land  ? 

10.  Squared  paper  is  very  useful  for  making  plans  of  houses  and  other  objects. 

Draw  a  plan  of  the  first  floor  of  your  home  on  squared  paper,  taking  one  small  divi- 
sion (yV  inch)  to  represent  one  foot  in  the  actual  house. 

11.  Squared  paper  may  be  used  to  draw  maps  by  measuring  the  distances  to  im- 
portant points  from  two  side  lines  at  right  angles  to  each  other. 

Draw  a  map  of  your  school  grounds  on  squared  paper,  taking  one  small  division 
(TV  inch)  to  represent  five  or  ten  feet,  as  is  convenient.  Mark  all  trees  and  buildings 
by  measuring  the  distance  from  each  of  them  to  the  front  sidewalk  and  to  the  side  line 
of  the  lot. 

12.  We  frequently  wish  to  find  the  areas  of  very  irregular  objects  that  occur  in 
nature.    Thus  the  areas  of  leaves  are  important  in  agriculture,  since  the  amount  of 
growing  power  of  a  plant  depends  on  the  area  of  its  leaves. 

Press  an  irregular  leaf  on  squared  paper,  and  determine  its  area  after  tracing  its 
edge  in  pencil. 

13.  Stick  two  pins  firmly  in  a  sheet  of  squared  paper,  about  one  inch  apart. 
Around  them  tie  loosely  a  loop  of  stout  thread  about  three  inches  long.     Stretch  this 
loop  taut  with  the  point  of  your  pencil,  and  move  the  pencil  around. 

The  curve  formed  is  called  an  ellipse.    Find  its  area. 


CONSTRUCTIVE  GEOMETRY 


61 


MISCELLANEOUS  APPLICATIONS 

i.   Draw  a  half  circle.     Draw  two  smaller  half  circles  whose  diameters  are  the 


B         D          A          E        C 
FIG.  32 

two  radii  of  the  larger  circle. 

This  figure  is  used  as  the  basis  of  many  ornamental  designs. 

2.   Copy  accurately  each  of  the  following  designs  enlarged  in  the  ratio  1:4.     These 
designs  are  all  based  on  the  construction  of  Ex.  i. 


DOUBLE  SCROLL 


FIG.  33 

3.  Draw  a  regular  octagon  (Ex.  9,  p.  51),  and  draw  all  the  possible  diagonals. 
How  many  are  there  ? 

4.  Draw  a  polygon  of  sixteen  equal  sides  inscribed  in  a  circle.     Draw  all  the 
possible  diagonals. 

A  favorite  test  of  technical  skill  in  using  drawing  instruments  is  to  draw  on  a 
large  sheet  of  paper  a  polygon  of  sixty-four  equal  sides  inscribed  in  a  circle,  and  to 
draw  all  its  diagonals.  If  this  figure  is  attempted  at  all,  a  long  time  should  be  allowed 
for  its  completion,  since  there  are  1952  diagonals. 

In  general,  each  corner  of  a  polygon  can  be  connected  by  a  diagonal  to  all  but 
three  of  the  corners,  —  itself  and  the  two  nearest  it. 


62  CONSTRUCTIVE  GEOMETRY 

5.  To  find  the  center  of  a  given  circle  whose  center  is  unknown. 

(a)  Draw  a  circle  (or  a  portion  of  one),  keeping  the  center  unmarked  by  putting 
a  small  piece  of  pasteboard  under  the  compass  point. 

(b}  Mark  any  points  A,  B,  C  on  the  circle,  and  draw  the  lines  AB  and  BC. 

(c)  Draw  a  line  perpendicular  to  AB  at  its  middle  point.    Also  draw  a  line  per- 
pendicular to  BC  at  its  middle  point. 

(d)  Extend  these  two  perpendiculars  to  meet  at  a  point  0. 
This  point  O  is  the  center  of  the  circle. 

6.  Given  three  points  A,  B,  C  in  the  plane,  draw  a  circle  through  them.     Do  not 
put  A,  B,  C  in  the  same  straight  line. 

7.  Draw  a  triangle  of  any-  form,  and  draw  a  circle  that  passes  through  its  three 
corners. 

Such  a  circle  is  called  a  circumscribed  circle. 

8.  To  round  off  a  sharp  corner  by  a  circle  touching  both  sides  of  an  angle. 

(a)  Draw  any  angle  ABC. 

(b)  Divide  the  angle  into  two  equal  parts.     (See  Ex.  7,  p.  32.) 

(c)  From  any  point  P  in  the  dividing  line,  draw  a  perpendicular  PD  to  one  of 
the  sides  of  the  angle  BC,  meeting  that  side  at  a  point  D. 

(d~)  About  P  as  center,  with  a  radius  equal  to  the  perpendicular  PD,  draw  a 
circle. 

9.  A  street  car  line  turns  a  corner  at  which  two  streets  are  perpendicular.     To  turn 
a  circle  of  20  feet  radius  is  inserted.     Draw  a  diagram  of  the  track/  if  the  width  of  the 
track  is  4  feet. 

Such  designs  for  street  car  lines  and  railroads  are  sometimes  very  complex. 
Cases  in  which  the  angle  between  LN  and  NQ  is  not  a  right  angle,  and  cases  in  which 
there  are  two  or  more  turns,  occur  frequently.  Additional  figures  of  this  kind  may  be 
made  if  there  is  time  for  it. 


\9-20  ft4,p 


M         IN 

FIG.  34 


10.   Draw  any  triangle  and  divide  each  of  the  angles  into  two  equal  parts. 

These  three  dividing  lines  meet  in  a  point. 

With  this  point  as  center,  draw  a  circle  that  touches  each  side  of  the  triangle. 


CONSTRUCTIVE  GEOMETRY  65 

11.  The  belt  of  a  sewing  machine  runs  over  two  wheels  whose  centers  are  18  inches 
apart.     The  diameters  of  the  wheels  are  1 2  inches  and  4  inches  respectively. 

Draw  this  figure  to  scale.  The  parts  of  the  belt  between  the  wheels  can  be  drawn 
by  placing  the  ruler  so  as  to  touch  both  circles. 

Measure  in  degrees  with  the  protractor  the  portion  of  the  surface  of  each  wheel 
in  contact  with  the  belt. 

12.  Draw  the  following  patterns  enlarged  i :  4.     Explain  how  each  one  is  drawn. 


FIG.  35 


13.   Copy  the  following  ornamental  designs  for  Gothic  windows. 


B         A 


14.  Light  is  reflected  from  a  mirror  so  that  the  reflected  ray  makes  the  same  angle 
with  the  mirror  that  the  original  ray  makes.     Copy  this  figure. 


LIGHT 


FIG.  37 


15.  Draw  a  figure  to  represent  two  mirrors  that  stand  at  an  angle  of  45°  and 
show  that  a  ray  of  light  which  strikes  one  of  them  parallel  to  the  other  is  reflected  exactly 
to  its  source. 

1 6.  Two  mirrors  stand  at  an  angle  of  60°.     Draw  a  figure  to  show  how  a  ray  of 
light  is  reflected  which  strikes  one  of  these  mirrors  parallel  to  the  other  one. 

Figures  may  be  drawn  to  illustrate  the  following  principle :  Any  point  of  an  object 
and  its  image  in  a  mirror  are  equally  distant  from  the  mirror,  and  the  line  joining 
object  and  image  is  perpendicular  to  the  mirror. 


66 


CONSTRUCTIVE  GEOMETRY 


17.   The  following  table  shows  the  notes  taken  by  a  surveyor  in  surveying  a 
broken  line  ABCDEF.     Draw  a  map  of  this  to  scale. 


FIG.  38 


Station 

A  (to  B) 

B  (to  C) 

C  (to  D) 

D  (to  E) 

J2,(to  F) 

Direction 

N 

N35°E 

N8o°E 

E 

56o°E 

Distance 

150  ft. 

75ft. 

100  ft. 

125  ft. 

200  ft. 

N  35°  E  means  35°  to  the  east  of  true  north. 
S  60°  E  means  60°  to  the  east  of  true  south. 
Measure  the  distance  and  the  direction  from  A  to  each  of  the  points  C,  D,  E,  F. 

18.  These  are  the  notes  for  a  railway  switch,  beginning  at  the  point  where  the 
switch  is  to  leave  the  main  line.  Lay  out  a  map  of  the  switch,  using  ruler  and  pro- 
tractor, to  the  scale  of  1000  feet  to  the  inch. 


Stations 

i 

2 

3 

4 

5 

6 

Bearings 

N2o°E 

N40°E 

N45°E 

N6o°E 

E 

Ei5°S 

Distances 

500  ft. 

1250  ft. 

IOOO  ft. 

20OO  ft. 

500  ft. 

IOOO  ft. 

Surveying  is  often  carried  on  in  the  manner  described  on  this  page.  The  surveyor 
ordinarily  uses  a  telescope  mounted  on  a  tripod  to  measure  the  angles  accurately. 
Reasonably  good  results  can  be  found,  however,  without  any  special  instruments,  by 
the  simple  expedient  of  sighting  across  pins  stuck  in  a  level  board,  and  arranging  the 
pins  so  that  the  line  of  sight  from  the  middle  pin  across  either  one  of  two  others  passes 
through  a  point  located  at  one  of  the  two  corners  of  the  route  of  the  survey.  A  group 
of  students  can  readily  survey  in  this  manner  the  bank  of  some  stream  or  the  track  of 
some  railroad,  and  then  draw  a  map  of  it. 

Ships  are  often  navigated  on  short  trips  by  a  similar  scheme,  keeping  the  direction 
by  means  of  the  ship's  compass,  and  sailing  in  straight  lines  as  long  as  is  convenient. 


CONSTRUCTIVE  GEOMETRY  71 

19.  An  egg-shaped  drainage  channel  as  shown  in  Fig.  39  is  formed  by  four  circular 
arcs.  The  circles  ABC  and  DEF  touch  in  G  and  the  circular  arcs  AF  and  CD  touch 
both  circles,  AC  being  a  diameter  of  the  larger  circle.  Make  a  copy  of  the  figure  to 
represent  the  case  hi  which  the  radii  of  ABC  and  DEF  are  2  feet  and  i  foot  respec- 
tively, choosing  the  centers  of  the  circles  AF  and  CD  by  trial,  somewhere  on  AC 
extended. 


FIG.  30 

[To  locate  the  center  of  AF,  for  example,  accurately,  proceed  as  follows.  Denote  by  M  and  N, 
respectively,  the  centers  of  the  circles  ABC  and  DEF.  Connect  N  to  a  point  P  on  AC  with  AP  =  GN. 
Draw  NX  so  that  Z  PNX  =  Z  MPN.  The  true  center  of  AF  is  the  intersection  of  AC  and  NX.} 

20.  Make  a  copy  of  the  adjoining  figure,  which  represents  a  siphon  carrying  water 
from  one  vessel  to  another. 


FIG.  40 

21.   Make  a  copy  of  the  adjoining  figure,  which  represents  the  outlines  of  a  steam 
engine. 


FIG.  41 


A  variety  of  geometric  outline  drawings  of  engines  can  be  found  in  encyclopedias, 
books  on  engines,  and  even  in  advertisements.  The  student  may  discover  such  a 
drawing  and  copy  it. 


72  CONSTRUCTIVE  GEOMETRY 

22.   Make  a  copy  of  the  adjoining  figure,  which  represents  the  action  of  a  magic 
lantern  in  throwing  a  picture  on  a  screen. 


FIG.  42 

Textbooks  on  physics,  and  those  on  geometrical  optics,  contain  a  great  variety 
of  figures  of  this  sort.  The  action  of  lenses,  cameras,  telescopes,  microscopes,  the 
human  eye,  field  glasses,  etc.,  can  be  illustrated  vividly  by  such  figures. 

23.  Make  a  copy  of  the  following  figure,  which  represents  the  action  of  a  force- 
pump. 


FIG.  43 

The  circular  top  of  the  equalizing  tank  /  may  be  drawn  accurately  by  means  of 
Ex.  8,  p.  62,  so  that  there  will  be  no  break  in  the  smoothness  of  the  surface  where  it 
joins  the  straight  sides.  The  purpose  of  this  tank  is  to  equalize  the  flow  by  means  of 
the  varying  compression  of  the  air  in  the  tank. 


CONSTRUCTIVE  GEOMETRY 


75 


24.   Make  a  copy  of  the  adjoining  figure  which  represents  the  outlines  of  an  elec- 
tric dynamo. 


25.  Draw  on  a  larger  scale  the  following  diagrams.  If  these  are  cut  out  of  paper 
and  folded  along  the  dotted  lines  they  can  be  closed  into  solid  figures  called  the  regular 
solids. 


FIG.  45 


In  actually  making  such  models,  little  flaps  may  be  left  attached  to  the  edges 
which  are  to  be  joined  together  later.  Such  flaps  are  very  convenient  in  pasting  the 
figures  together  to  form  the  solids. 


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